On GMRES-Equivalent Bounded Operators
Leonid Aronovich Knizhnerman · SIAM Journal on Matrix Analysis and Applications · 2000
Given a bounded linear operator A in a Hilbert space $\calH$ and a nonzero vector $\mbox{\eufm r}\in\calH$, we construct a unitary operator U and (under some conditions) bounded self-adjoint operators P and T (nonnegative definite and indefinite, respectively) such that all the residual Krylov subspaces of $(A,\mbox{\eufm r})$, $(U,\mbox{\eufm r})$, $(P,\mbox{\eufm r})$, and $(T,\mbox{\eufm r})$ of the same dimension for the equation $Ax=\mbox{\eufm r}$ are equal. When possible (for example, for U and P, provided 0 is outside the field of values of A), we estimate a gap in the spectrum of U and the condition numbers of P and T. Some attainability results are also established. It is shown that some analogous matrix assertions are valid, which can be obtained by means of degenerating the operator case. Numerical examples are presented for the finite-dimensional case.